Second order partial derivatives, the Taylor series and the degenerate case.

Hello!

I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived:

and are the first and second partial derivatives of a, respectively.

The first few terms in the Taylor series (those that appear above), can be used to make an approximation of the change in function. At a critical point:

And so we are left with:

Which is a quadratic approximation of the change in the function at a critical point.
For me, its easier to understand what is happening by factoring out y:

From this, the discriminant can be used to determine local min, local max, saddle etc properties. If the discriminant, , equals zero. Then it is not possible to tell what is happening. This is because higher order terms are then important.